\(2{\log _{x-2}}\sqrt 3 + {\left( {x-4} \right)^2}{\log _3}\left( {x-2} \right) = {\left( {x-4} \right)^2}{\log _{x-2}}3 + 2{\log _3}\sqrt {x-2} .\)
Запишем ОДЗ:
\(\left\{ {\begin{array}{*{20}{c}}{x-2 > 0,}\\{x-2 \ne 1}\end{array}\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x > 2,}\\{x \ne 3}\end{array}\,\,\,\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,x\, \in \,\left( {2;3} \right) \cup \left( {3;\infty } \right).} \right.} \right.\)
\({\left( {x-4} \right)^2}{\log _3}\left( {x-2} \right)-{\left( {x-4} \right)^2}{\log _{x-2}}3 + {\log _{x-2}}3-{\log _3}\left( {x-2} \right) = 0\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,{\left( {x-4} \right)^2}\left( {{{\log }_3}\left( {x-2} \right)-{{\log }_{x-2}}3} \right)-\left( {{{\log }_3}\left( {x-2} \right)-{{\log }_{x-2}}3} \right) = 0\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,\left( {{{\left( {x-4} \right)}^2}-1} \right)\left( {{{\log }_3}\left( {x-2} \right)-\dfrac{1}{{{{\log }_3}\left( {x-2} \right)}}} \right) = 0\,\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{{{\left( {x-4} \right)}^2}-1 = 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{{{\log }_3}\left( {x-2} \right)-\dfrac{1}{{{{\log }_3}\left( {x-2} \right)}} = 0}\end{array}\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x-4 = 1,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{x-4 = -1,\,\,\,\,\,\,\,\,\,}\\{\log _3^2\left( {x-2} \right) = 1}\end{array}} \right.} \right.\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = 5,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{x = 3,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{{{\log }_3}\left( {x-2} \right) = 1,\,\,}\\{{{\log }_3}\left( {x-2} \right) = -1}\end{array}} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = 5,\,\,}\\{x = 3,\,\,}\\{x = 5,\,\,}\\{x = \dfrac{7}{3}.}\end{array}} \right.\)
Корень \(x = 3\) не удовлетворяет ОДЗ.
Ответ: \(\dfrac{7}{3};\,\,\,5\).